Об особенностях вычисления производных высших порядков для идентификации формы графических объектов
Methods of calculation of high order derivatives are considered on a basis: interpolation formulas; “without difference methods of calculation of derivatives”; applications of convolution with replacement of differentiation by integration operation; differentiation with use of quadratures on C. Lanczos; the method of Numerova. The comparative analysis of methods of calculation of high order derivatives on accuracy of calculations with use as the sample of the derivatives calculated in package Maple with 20 digit decimal accuracy is carried out. It is shown that all methods are almost equivalent on accuracy and are reduced to convolution calculation between differentiated function and some window which coefficient depend on an applied method. For carrying out of experiments the special program complex is developed for calculation of high order derivative (up to 7th) the tabulated functions with various step. Grids with steps from 0.005 to 0.1 have been investigated. Irrespective of a method of calculation of derivatives it has been defined that optimum value of step mesh for 64 digit arithmetic’s the step is from 0.01 till 0.05. Value of smooth functions differs less than their accuracy of representation at smaller value of a step, and at greater step - the differentiation error increases. Results of experiments confirm N.N. Kalitkin’s theoretical conclusions.